What is the integrating factor for the equation dy/dx + 2y = 6?

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Question: What is the integrating factor for the equation dy/dx + 2y = 6?

Options:

Correct Answer: e^(2x)

Solution:

The integrating factor is e^(∫2dx) = e^(2x).

What is the integrating factor for the equation dy/dx + 2y = 6?

What is the integrating factor for the equation dy/dx + 2y = 6?

Step 1: Identify the standard form of the differential equation, which is dy/dx + P(x)y = Q(x). In this case, P(x) = 2 and Q(x) = 6.
Step 2: Find the integrating factor, which is given by the formula e^(∫P(x)dx). Here, P(x) = 2.
Step 3: Calculate the integral of P(x): ∫2dx = 2x.
Step 4: Substitute the result back into the formula for the integrating factor: e^(∫2dx) = e^(2x).
Step 5: Conclude that the integrating factor for the equation dy/dx + 2y = 6 is e^(2x).

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